He is the Avataar of Shesh Naag. See also Balaraam - he is also said to be the Avataar of Shesh Naag

There have been several Avataar of several gods on this Prithvi. It is not only Vishnu but many other gods also have taken Avataar on this Prithvi. One such Avataar is of Lakshman. Lakshman came with Shree Raam and is the Avataar of Aadi Shesh Jee.

He was the son of Dasharath and Sumitra. he had a brother named Shatrughn. Lakshman was married to Urmilaa, the younger sister of Seetaa. When Raam was sent in exile by their step-mother Kaikeyee, Lakshman also went with Him. he was always helpful to Raam. He could not live without Raam.

It is said that he had won sleep, so till they were in forest he never slept and always kept vigil of Raam and Seetaa. If Seetaa had not said to him odd things, he would have never left Seetaa alone in the Aashram and Raavan had no chance to abduct Her.

Vaalmeeki Raamaayan says - when Raam came on Rishyamook Parvat and told about His problem to Sugreev, he showed them cloth and some jewelry of some woman going crying by skyway. Raam showed them to Lakshman and asked him - "Do you recognize it? is this earring of Seetaa?" Lakshman replied - "Bhaiyaa, I cannot recognize her earrings because I always worshipped Her feet only, I never saw Her face."

What Urmilaa did without Lakshman? It is said that she asked sleep and she slept all the time till Lakshman lived in forest. There is long poetry written by Maithilee Sharan Gupt "Urmilaa" in which the poet has successfully painted the picture of Urmilaa's feelings.

There are four Yug - Aadi Shesh or Anant has taken different Avataar in different Yug
"Anantah Prathamam Roopam, Tretaya Lakshmanascha Tathah
Dwaapare Balabhadrashcha Kalau Kashchit Bhavishyati"

The meaning of this Shlok -
"In Krit Yug (1st Yug) Aaadi Shesh was the Snake Bed of the Lord
In Tretaa Yug (2nd Yug) he was incarnated as Lakshmana, brother of Raam
In Dwaapar Yug (3rd Yug) he was incarnated as Balaraam, brother of Krishn, and
In Kali Yug (4th Yug) he was incarnated as Swami Ramanuja,
From the above verses it is clear that Swami Raamaanuj is the Avataar of Aadi Shesh